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Bending of Light by Gravity

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Principles of Astrophysics

Part of the book series: Undergraduate Lecture Notes in Physics ((ULNP))

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Abstract

To this point we have examined how massive objects move under the influence of gravity. Einstein taught us that light’s motion is affected by gravity as well. Despite being relativistic, gravitational light bending can be studied with a quasi-Newtonian framework to obtain a new way to probe mass in the universe.

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Notes

  1. 1.

    This is related to the analysis in Sect. 3.1, but now applied to an unbound orbit.

  2. 2.

    You might wonder whether it makes sense to take the limit of the gravitational force as m → 0, but in general relativity we learn that energy gravitates.

  3. 3.

    Gravitational lensing by black holes does require a full relativistic treatment (see [2] for a review).

  4. 4.

    “Strong” is a relative term; the bending angle is still in the small-angle regime.

  5. 5.

    Having the image subtend exactly the same azimuthal angle as the source requires a radial deflection and thus is limited to circular lenses. The concept of tangential stretching is general, though.

  6. 6.

    You may be familiar with the principle of least time, but local minima are not the only stationary points. As a function of two dimensions, τ can also have local maxima and saddle points.

  7. 7.

    Don’t blame me—I didn’t invent the names! For the record, “WIMP” was introduced first, and “MACHO” was chosen deliberately (see [8]).

  8. 8.

    By contrast, variable stars tend to change color as they change brightness.

  9. 9.

    The SIS model can also be expressed in terms of the velocity dispersion, which is \(\sigma = v_{c}/\sqrt{2}\).

  10. 10.

    The shear is basically a tidal effect analogous to what we studied in Chap. 5.

  11. 11.

    “External” because it comes from outside the main lens galaxy (i.e., from the neighbors). Note that we drop the subscript on γ to simplify the notation.

  12. 12.

    An alternative approach is to make as few assumptions as possible (although assumptions can never be avoided altogether), and then deal with the large range of mass models that are consistent with the observed images [17].

  13. 13.

    We hope. Correlations among the intrinsic shapes of galaxies could present a challenge for weak lensing [20, 21], but they are generally expected to be small and there are ways to deal with them in a weak lensing analysis [22].

  14. 14.

    This is not strictly a cone because the edge is not straight, but the terminology is helpful because the region does grow with distance behind the black hole.

References

  1. C.R. Keeton, A.O. Petters, Phys. Rev. D 72(10), 104006 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  2. V. Bozza, Gen. Relativ. Gravit. 42, 2269 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. F.W. Dyson, A.S. Eddington, C. Davidson, R. Soc. Lond. Philos. Trans. Ser. A 220, 291 (1920)

    Article  ADS  Google Scholar 

  4. W.W. Campbell, Publ. Astron. Soc. Pac. 35, 11 (1923)

    Article  ADS  Google Scholar 

  5. C.R. Keeton, S. Burles, P.L. Schechter, J. Wambsganss, Astrophys. J. 639, 1 (2006)

    Article  ADS  Google Scholar 

  6. S. Dong et al., Astrophys. J. 642, 842 (2006)

    Article  ADS  Google Scholar 

  7. A. Gould et al., Astrophys. J. 720, 1073 (2010)

    Article  ADS  Google Scholar 

  8. K. Griest, Astrophys. J. 366, 412 (1991)

    Article  ADS  Google Scholar 

  9. J.L. Feng, Ann. Rev. Astron. Astrophys. 48, 495 (2010)

    Article  ADS  Google Scholar 

  10. C. Alcock et al., Astrophys. J. 541, 734 (2000)

    Article  ADS  Google Scholar 

  11. C. Alcock et al., Astrophys. J. 542, 281 (2000)

    Article  ADS  Google Scholar 

  12. C. Alcock et al., Astrophys. J. 479, 119 (1997)

    Article  ADS  Google Scholar 

  13. D. Kubas et al., Astron. Astrophys. 435, 941 (2005)

    Article  ADS  Google Scholar 

  14. B.S. Gaudi, Ann. Rev. Astron. Astrophys. 50, 411 (2012)

    Article  ADS  Google Scholar 

  15. J.P. Beaulieu et al., Nature 439, 437 (2006)

    Article  ADS  Google Scholar 

  16. C.R. Keeton, C.S. Kochanek, Astrophys. J. 495, 157 (1998)

    Article  ADS  Google Scholar 

  17. P. Saha, L.L.R. Williams, Astron. J. 127, 2604 (2004)

    Article  ADS  Google Scholar 

  18. T. Treu, Ann. Rev. Astron. Astrophys. 48, 87 (2010)

    Article  ADS  Google Scholar 

  19. K.C. Wong, C.R. Keeton, K.A. Williams, I.G. Momcheva, A.I. Zabludoff, Astrophys. J. 726, 84 (2011)

    Article  ADS  Google Scholar 

  20. R.A.C. Croft, C.A. Metzler, Astrophys. J. 545, 561 (2000)

    Article  ADS  Google Scholar 

  21. A. Heavens, A. Refregier, C. Heymans, Mon. Not. R. Astron. Soc. 319, 649 (2000)

    Article  ADS  Google Scholar 

  22. J. Blazek, R. Mandelbaum, U. Seljak, R. Nakajima, J. Cosmol. Astropart. Phys. 5, 041 (2012)

    Article  ADS  Google Scholar 

  23. N. Kaiser, G. Squires, Astrophys. J. 404, 441 (1993)

    Article  ADS  Google Scholar 

  24. J.P. Kneib, P. Natarajan, Astron. Astrophys. Rev. 19, 47 (2011)

    Article  ADS  Google Scholar 

  25. D. Clowe, M. Bradač, A.H. Gonzalez, M. Markevitch, S.W. Randall, C. Jones, D. Zaritsky, Astrophys. J. Lett. 648, L109 (2006)

    Article  ADS  Google Scholar 

  26. M. Bradač, S.W. Allen, T. Treu, H. Ebeling, R. Massey, R.G. Morris, A. von der Linden, D. Applegate, Astrophys. J. 687, 959 (2008)

    Article  ADS  Google Scholar 

  27. B. Famaey, S.S. McGaugh, Living Rev. Relativ. 15, 10 (2012)

    Article  ADS  Google Scholar 

  28. A. Refregier, Ann. Rev. Astron. Astrophys. 41, 645 (2003)

    Article  ADS  Google Scholar 

  29. A. Albrecht et al., ArXiv e-prints arXiv:astro-ph/0609591 (2006)

    Google Scholar 

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Authors and Affiliations

  1. Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, USA

    Charles Keeton

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Keeton, C. (2014). Bending of Light by Gravity. In: Principles of Astrophysics. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9236-8_9

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  • DOI: https://doi.org/10.1007/978-1-4614-9236-8_9

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